As a reminder, the opinions expressed on my website are mine and mine alone, and do not represent those of my employers or institution.

If you’ve talked to me in the past few years, you’ll certainly know how I feel about Silicon Valley and the way generative AI has been developed. This is not a post about that, on that subject, I will leave you with this study from 2024 on the effects generative AI expansion may have on our planet as well as this paper, which explains how recent developments in big tech may lead to illiberalism, and we will be done with that.

Instead, this is a post about my experiences using generative AI for mathematics: it is the perspective of a mathematical researcher whose field can lead slightly on the side of “more abstract,” and the perspective of someone who certainly doesn’t find joy in using machines (I had a flip phone until 2017, and frequently wish I still did), but does take joy in new discoveries, and has a desire to stay in academia. At the University of Georgia, we have had free access to a form of Google Gemini Pro for the past year, which to my understanding was slightly limited but was essentially close to frontier-level close to a year ago (according to FrontierMath, see the Tier 4 v1 results in November 2025). At the start of last year, I fell somewhere between the lines of “wanting absolutely nothing to do with it” and “trying it out of morbid curiosity.” At the time, the problems I was working on had little to no use for generative AI and fell somewhere on the more abstract end. I had tried using Gemini as a spell-and-sanity checker, but even then it appeared to miss many typos or small errors, and frequently flagging correct data. However, I did get some use from it for literature review, either finding papers that would serve as good background for what I was working on or for finding niche results.

However, mathematically, the answers to any mathematical questions I posed were predominantly useless. Most were clearly ignorable from the start, I have one recollection of a statement posed that seemed viable, to which I actually felt quite excited (and scared), but after a full week, realized that it couldn’t possibly work, and actually found a proof of the fact. I was curious of what the generative AI’s response would be when I gave it the proof, and to my surprise, it replied admitted defeat. Perhaps this was sycophancy at work, but if so, it worked on me, as it did give me a sense that there was “learning” in progress.

Things started to change midway through the spring semester. There were plenty of times while writing a preprint classifying endotrivial complexes over an arbitrary commutative Noetherian ring with Omar Gomez that I found Gemini useful. In this case, it was to ask fairly elementary questions any ring theorist could answer: whether x or y property was something any relevant ring satisfies. Generally, it would quickly spit out an easy counterexample, which I imagine saved us hours. Additionally, there was one minor point it brought up in the homotopy theory section of the paper, and while the proof(s) it gave were all wildly incorrect, the point ended up being true for different reasons. At this point, my attitude shifted towards “these things can be useful if one poses an elementary enough question.”

Then while in Santa Cruz visiting some friends in May, I ran into a friend and another postdoc, Nic Brody, and we had a good long chat about LLMs. He was more optimistic and scared, I was more pessimistic but also scared. The next day, I received an email from him with a link to OpenAI’s unit distance theorem disproof. What timing…

From there, the floodgates opened. I spent much of the summer traveling around doing math and relaxing with friends and family (see my travel page for the math-related stuff). Every week, I would see some new absurd counterexample posted. Meanwhile, Gemini, whenever I would query it, didn’t seem to get much better. I hadn’t realized just how much of a gap there was between Google’s products and OpenAI’s and Anthropic’s. On the other hand, I mostly disregarded all the “sandbox breakout” press releases and the buzz around Mythos being “too powerful,” writing it off (and still writing it off now mostly) as a twisted form of AI hype. But eventually, after seeing where Fable stood on the FrontierMath leaderboard, in August I purchased an account. Perhaps it was out of fear and anxiety, and perhaps it was out of curiosity, and perhaps the PR from all the disproofs did its work on me.

There was a pretty immediate difference in ability; while the generative AI would still frequently generate something that was, well, wrong, prompting “double check everything you’ve just said” over and over before actually looking to verify things myself would clear things up pretty well, as incorrect statements were sometimes retracted. Sometimes on the other hand, the model would exhibit the same behavior as Gemini, going in circles on an incorrect statement, missing the point. Nevertheless, the first major improvement I saw was not actually in counterexample creation (that was to come), but rather being able to show fairly abstract (although not terribly difficult) lemmata, something Gemini had certainly been failing me on with things outside of ring theory.

While doing revisions with Timothy De Deyn on our preprint on classification of thick tensor ideals in noncommutative tensor-triangular geometry, a question was raised by the referee if a non-example of a property called “compact detection” existed. This is something we’d briefly thought about but figured wasn’t worth our time. On a whim, I asked Fable, and it spit out a nice counterexample based on the free monoid of 2 elements, as well as another counterexample in the paper we’d cooked up earlier. I was impressed; I asked it if this (extremely contrived) example was found anywhere in the literature and it replied no, it was motivated from our prior example. I still have my doubts on that, and figure that the non-categorified version must be a known example in semigroup theory, but still. This was the first taste of counterexample generation.

From there, I decided to test it - there was a lingering question (see section 8) from Omar and I’s preprint about orientability of representation spheres. I’d thought about this question somewhat and had a proof of a classification of orientables for 2-groups. I told Fable this and told it to prove that one could use this to lift the question to all groups (as the only groups that are relevant are those of even order). It lifted the question to groups with a normal 2-Sylow, something I’d already sketched out, and drew together a few other equivalent conditions, which I hadn’t considered, but in hindsight were pretty obvious. But after more prodding, it found a counterexample to my claim on a group of order 112! I was quite surprised - this was not discovered by brute-force GAP computation (which in hindsight, I should have tried first), but rather proven purely conceptually. In fact, the proof is rather nice, although it pretty clearly was borrowing ideas from elsewhere.

From there, I posed it another question, the remaining open question (Remark 4.2) from a paper written with Nadia Mazza. In this case, I told Fable to find another counterexample, and instead, it spit out a (very flawed) proof sketch that no such counterexample exists. I don’t know if the proof sketch could be saved, but the general idea was to use the existence of so-called characteristic idempotents of fusion systems, to which I recalled that my advisor, Robert Boltje, had also suggested looking at these a while back. After reading a paper or two on this, I realized that there was a very clean answer to the problem using biset functor formality. The resulting preprint is now on arXiv and submitted - all in all, there are maybe about 2 pages of actual work, and the rest is background or easy.

Turning back to the orientability question, I had Claude Opus design some GAP code to check for other examples of counterexamples, and to test for some other phenomena. This was perhaps where I was most impressed; the model wrote and modified GAP code that far exceeded my expectations in terms of computational efficiency. Although perhaps this was to be expected: LLMs became highly proficient at writing code (although its efficiency was perhaps questionable) well before they became proficient at math. It was able to run some nontrivial character-theoretic checks on sporadic simple groups, which are, well, large. This led to a few other related counterexamples (which were at this point expected), but one property I found no such counterexamples to. After thinking on the question myself for a few days, as well as querying Fable, I concluded the problem was difficult, stated it as a conjecture, and finished up writing. The preprint should be available on arXiv very soon.

All and all I’m still not sure what to think. First of all, the questions I asked were highly computational in nature relative to my normal work. And the results, looking back, are not terribly complicated; in the end almost all of the math I did myself (the main exceptions being the GAP code and the counterexample). I still find myself agreeing with Tobias J. Osborne’s take that while LLMs and reasoning models are great at finding counterexamples and specific results, they probably won’t be good at theory development or complex multi-stage proofs requiring new constructions for a while. But I have been wrong before; my timeline for these things was way off, as were my predictions for the accuracy timeline. And I still think that while these things are trained on human math, they won’t exceed a human mathematician in ways beyond raw compute power and literature searching, i.e., finding connections, but those already are very powerful tools to have. I cannot deny that these tools are already considerably useful and as an aspiring mathematician, feel almost compelled to use them. I mean, I just wrote two (fairly short, insignificant, and computation-heavy) preprints in two weeks! Without the AIs writing anything! That boggles the mind.

At a very minimum, while I don’t think the models will be able to assist me with the harder questions I’m tackling right now, from an efficiency standpoint it is at least worth querying them to see if they can spit out a neat counterexample and save me time. The postdoc stage is the time for production, and the job market is worse than ever. But I cannot help but think about the joy that comes with exploration and discovery that is lost by consulting the Library of Babel, the great Stochaistic Parrot. Before school started, I went on a weeklong canoeing trip with a good friend in Woodland-Caribou Provincial Park in Ontario, Canada. During this isolation from technology and society, I fell into a wonderful state of simplicity, of not worrying about what I had to do next, but simply paddling, walking, sitting, and enjoying where I was at that point, not having so many distractions and responsibilities pulling at my attention. Mind you, these trips are also very physically demanding, which also serves as a good distraction. When I can be in a similar state practicing mathematics, I find it the most blissful of times. On the flip side, when I feel like I am only doing mathematics when I need to for my career’s sake, it causes me emotional and possibly physical pain. And the points I laid out in the intro (which I promised I’d not elaborate on) cause me those things as well. So I ultimately don’t know what to do.

I leave you with the writing from my former professors Jon Jacobsen, In Praise of Acoustic Mathematics, and Francis Su, A Meditation on Mathematical Joy.

  • sam